Let $X=Spec A$ be an affine scheme and $Y\subseteq X$ a closed subscheme with unclusion $i:Y\hookrightarrow X$ defined by a coherent ideal $\mathscr{I}$. For a $\mathscr{O}_{X}$-module $\mathscr{F}$ the sheaf $\Gamma_{Y}\mathscr{F}$ is the kernel of the canonical map $\mathscr{F}\rightarrow j_{*}j^{-1}\mathscr{F}$, where $j:X\backslash Y\hookrightarrow Y$ is the canonical inclusion of the open complement of $Y$ into $X$, i.e. $\Gamma_{Y}\mathscr{F}$ can be viewed as the subsheaf of $\mathscr{F}$ of sections whose support is contained in $Y$.
In a book I am reading the following argument is made:
If a local section $s\in \Gamma(U;i^{-1}\mathscr{F})$ ($U\subseteq Y$ open) satisfies $(i^{-1}\mathscr{I})s=0$, then $s$ lies in $\Gamma(U;i^{-1}\Gamma_{Y}\mathscr{F})$.
Now this looks rather weird to me. For if we take, say, the structure sheaf $\mathscr{O}_{X\backslash Y}$, then to me the extension $j_{!}\mathscr{F}$ by zero outside of $X\backslash Y$ looks very much like a $\mathscr{O}_{X}$-module the local section of which satisify $(i^{-1}\mathscr{I})s=0$. At the same time none of those local sections lie in $\Gamma(U;i^{-1}\Gamma_{Y}\mathscr{F})$ unless they are zero.
Are there any reasonable assumptions on $X,Y$ or $\mathscr{F}$ other that $X=Y$ and $\mathscr{F}=0$ that would make this argument work?
EDIT: It has been pointed out by Roland that this is not a counterexample. However I'd be glad if anyone could help me understand the argument or clarify under what conditions it works.